Colloquium: Illuminating the Kapitza-Dirac effect with electron matter optics

The observation of the Kapitza-Dirac effect raises conceptual, theoretical, and experimental questions. The Kapitza-Dirac effect is often described as diffraction of free electrons from a standing wave of light or stimulated Compton scattering. However, for the two-color Kapitza-Dirac effect these two interpretations appear to lead to paradoxical conclusions. The discussion of this paradox deepens our understanding of both of these versions of the Kapitza-Dirac effect.

Figure 1

(Color online) Matter-optics analogy. Diffraction of a light wave by a material grating is shown (left). A schematic of an electron matter wave diffracted by a laser beam, illustrating that the roles of the matter and light are reversed for the KD effect (right).

Figure 2

Standing-wave fields. The electric $\left(E_z\right)$ and magnetic $\left(B_y\right)$ fields of a standing wave and the force $\left(F_x\right)$ on an electron with velocity $\left(v_z\right)$ at position $x_1$ and $x_2$ are given. An electron placed at $x_1$ feels a Lorentz force in the same direction at times separated by half a light period, even though the fields have switched direction. This gives rise to a time-averaged potential from which electrons can diffract.

Figure 3

The position-momentum uncertainty relation (22; 40). Electrons moving from the left to the right through a laser focus are shown schematically. On the left, symmetric diffraction into many momentum states is shown; on the right, asymmetric Bragg scattering into two momentum states is shown. The diffracted electron beams are each separated by two photon momentum recoils $2\hslash k$. The diffractive regime can be tuned to the Bragg regime by reducing the laser-beam divergence from $\Delta {\varphi }_D$ to $\Delta {\varphi }_B$ with a less tight laser focus waist. The small uncertainty in the photon position for the diffractive case leads to a large uncertainty in the photon momentum (expressed in terms of $\Delta {\varphi }_D$), which allows many diffraction orders to be reached. See text for a detailed explanation.

Figure 4

The energy-time uncertainty relation. The momentum (in units of photon recoil) indicated along the horizontal axis versus kinetic energy of a free electron indicated along the vertical axis. In the Bragg regime (left) the interaction time is long and the associated energy uncertainty $\Delta E$ is much smaller than the recoil energy shift ${\delta }_B$. This explains why only the two momentum states can be reached for states incident at the Bragg angle. In the diffractive regime (right) the argument is reversed $(\Delta E⪢{\delta }_D)$.

Figure 5

(Color online) Wave and particle picture. The interaction between the electron beam and laser beam can be thought of as an electron wave experiencing a ponderomotive potential coinciding with a standing wave of light. The standing wave of light is the sum of two counterpropagating waves. Alternatively, we can think of the interaction as an electron undergoing a stimulated Compton scattering.

Figure 6

(Color online) Wave and particle picture. The interaction between the electron beam and a counterpropagating beam may be considered to be conceptually difficult. In the wave picture the beating wave does not create a standing wave of light. Curiously, in the particle picture both energy and momentum can be conserved in an up-conversion process. It appears likely that this process can take place (for a discussion see Sec. 2D).

Figure 7

The experimental setup used to observe the KD effect. A line is overlayed to schematically indicate the laser-beam paths.

Figure 8

Experimental data. The electron detection rate is presented as a function of detector position. Our data (solid dots) agree reasonably well with a numerical solution of the Schrödinger equation and clearly show the diffraction peaks that are the signature of the Kapitza-Dirac effect. The bottom figure shows the electron-beam profile with the laser beams turned off (bottom).

Figure 9

Bragg regime. The detection rate is given as a function of detector position. With the laser turned off (top) a symmetric beam profile is found, while the laser light (bottom) produces Bragg scattering.

Figure 10

Rocking curves. First-order (circles) and second-order (squares) Bragg scattering rates are shown as a function of the electron incident transverse momentum for the detector placed at the negative (open symbols) and positive (solid symbols) Bragg angles. The lines are a guide to the eye.

Figure 11

Two-color electron interaction simulation. Two counterpropagating laser beams can provide a time-averaged potential to an electron when their wavelengths ${\lambda }_1$ and ${\lambda }_2$ satisfy $n{\lambda }_1=m{\lambda }_2$, where $n$ and $m$ are integers. One wavelength is fixed at $1064\mathrm{nm}$ and the other is varied. Inset: The nonlinearity of the processes. The label $\omega \text{−}n\omega$ indicates the frequency of photons involved in the process. For example, the two-color interaction can involve the conversion of two photons of frequency $\omega$ into one with frequency $2\omega$.